Abstract: The solutions of the advective and the diffusion equation have been studies by means of accuracies of the first and second centereddifference approximations with the non-uniform grid. The effectscaaused by the variable mesh are as follows:First, leading to the difference of apparent wavelength among ten various grid-points for the same hamonic With a given wavelength.Second, producing an additional real component for the first centered difference, and attaching an imaginal component for the second centered difference approoimation when comparing with the corresponding approximation on the uniform grid.Those errors of the spatial finite-difference make the solution of advective equation to be a modulated and translated wave, and the solution of diffusion equation to carry with the false advective error. The remedies against the mentioned errors have been proposed with an artificial Viscosity in the advectiveecquation and with an additional abvection term in the aiffusion equation.